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% Correct Answers
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% Preface

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{\noindent
{\bf Math 108: Calculus for Business}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam 2}\quad{\sl October 11, 2000}\hfill
Section:\hbox to 2.5truein{\hrulefill}

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% Blank answer sheet

\vbox{\global\pageno=1
{\noindent
{\bf Math 108: Calculus for Business}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam 2}\quad{\sl October 11, 2000}\hfill
Section:\hbox to 2.5truein{\hrulefill}

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% Problems

\vbox{
%1

\noindent\llap{1.\ }%
(5 pts.) Assume that the marginal cost function of producing a certain
product is \break $C'(q)=3q^2-60q+1000$, and that the fixed cost is \$ 2000.
Find the total cost of producing 20 items.

\beginanswers
{(a)~%
18,000
\answerspace}
{(b)~%
16,000
\answerspace}
{(c)~%
12,000
\answerspace}
{(d)~%
15,000
\answerspace}
{(e)~%
13,000
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%2

\noindent\llap{2.\ }%
(5 pts.) You have recently invested \$ 10,000 in a money market account.
The account gains interest at an annual rate of 15\%, compounded
continuously. After how many years will your account value quadruple?

\beginanswers
{(a)~%
${\ln 2 \over .3}$
\answerspace}
{(b)~%
${2\ln 3 \over .15}$
\answerspace}
{(c)~%
${\ln 4 \over .15}$
\answerspace}
{(d)~%
8
\answerspace}
{(e)~%
12
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%3

\noindent\llap{3.\ }%
(5 pts.) Find the future value of a continuous income stream that flows
into a savings account at a constant rate of \$10,000 per year, that is
earning 8\% interest, and lasts 8 years.

\beginanswers
{(a)~%
$10,000(e^{-.64}-1)$
\answerspace}
{(b)~%
$10,000(e^{.64}-1)$
\answerspace}
{(c)~%
$125,000(e^{-.64}-1)$
\answerspace}
{(d)~%
$100,000$
\answerspace}
{(e)~%
$125,000(e^{.64}-1)$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%4

\noindent\llap{4.\ }%
(5 pts.) Find the present value of a perpetual income stream flowing
continuously at a rate of \$12,000 per year with interest rate of 8\% compounded
continuously.

\beginanswers
{(a)~%
150,000
\answerspace}
{(b)~%
100,000
\answerspace}
{(c)~%
125,000
\answerspace}
{(d)~%
120,000
\answerspace}
{(e)~%
133,333
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%5

\noindent\llap{5.\ }%
(5 pts.) Compute the improper integral:
$$
	\int^{\infty}_0 {{3x^2}\over {(x^3+1)^2}} dx.
$$

\beginanswers
{(a)~%
${1 \over 2}$
\answerspace}
{(b)~%
-1
\answerspace}
{(c)~%
1
\answerspace}
{(d)~%
The integral is divergent.
\answerspace}
{(e)~%
${-1\over 3}$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%6

\noindent\llap{6.\ }%
(5 pts.) Find the average value of $p(t)=t(t^2+1)^2$ over the interval $[0,2]$.

\beginanswers
{(a)~%
64
\answerspace}
{(b)~%
50
\answerspace}
{(c)~%
31/3
\answerspace}
{(d)~%
32
\answerspace}
{(e)~%
64/3
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%7

\noindent\llap{7.\ }%
(5 pts.) Find the constant $c$ such that the function 
$$
	p(t)={c\over {c+e^{-t}}}
$$
is a solution of the initial value problem
${dp \over dt}=p+p^2, p(0)=5.$

\beginanswers
{(a)~%
$c={-1 \over 2}$
\answerspace}
{(b)~%
$c={-5\over 4}$
\answerspace}
{(c)~%
$c={1 \over 2}$
\answerspace}
{(d)~%
$c=1$
\answerspace}
{(e)~%
$c=0$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%8

\noindent\llap{8.\ }%
(5 pts.) A person retires at age 68 with an IRA balance of \$1,000,000, and then makes continuous withdrawls at a rate of \$50,000 per year. Assuming an interest rate of 8\% compounded continuously, write a differential equation and an initial condition that describes the balance $M(t)$ in the IRA $t$ years after retirement.

\beginanswers
{(a)~%
${dM \over dt}=.08M+50,000; M(0)=1,000,000$
\answerspace}
{(b)~%
${dM \over dt}=.08M-50,000; M(0)=1,000,000$
\answerspace}
{(c)~%
${dM \over dt}=1,000,000-.08M; M(0)=50,000$
\answerspace}
{(d)~%
${dM \over dt}=50,000-.08M; M(0)=1,000,000$
\answerspace}
{(e)~%
${dM \over dt}=.08M+1,000,000; M(0)=50,000$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%9

\noindent\llap{9.\ }%
(5 pts.) Solve the initial value problem:
$$
	{dy \over dx} = 3x^2(y+1), y(1)=2.
$$


\vskip .2in

\beginanswers
{(a)~%
$y=e^{x^3-1}+1$
\answerspace}
{(b)~%
$y=3x^3$
\answerspace}
{(c)~%
$y=3e^{x^3-1}-1$
\answerspace}
{(d)~%
$y=e^{x^3+1}-1$
\answerspace}
{(e)~%
$y=3e^{x^3+1}-1$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%10

\noindent\llap{10.\ }%
(5 pts.) Deposits are made into an account in the following manner. An initial deposit of \$10,000 is made. After 5 years, additional money is deposited continuously into the account at a rate of \$5,000 per year. This lasts for 3 years. If the account pays 8\% compounded continously, how much money is in the account 8 years after the initial investment?

\beginanswers
{(a)~%
$10,000e^{.64}+\int_0^3 5,000e^{.08(3-t)}dt$
\answerspace}
{(b)~%
$10,000e^{.40}+\int_0^3 5,000e^{.08(3-t)}dt$
\answerspace}
{(c)~%
$10,000e^{.64}+\int_0^3 5,000e^{-.08t}dt$
\answerspace}
{(d)~%
$10,000e^{.40}+\int_0^3 5,000e^{-.08t}dt$
\answerspace}
{(e)~%
$5,000+\int^8_0 10,000e^{-.08}dt$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

% Appendix

{\vfil\eject\noindent
11. (20 pts.) Assume that for a certain item the demand curve is
$p=D(q)={90 \over {q+5}}$ and the supply curve is $p=S(q)=q+4$. 

\vskip .2in

a) Find the equilibrium price and the equilibrium quantity.

\vskip 1in

b) Find the consumer surplus.

\vskip 1.4in

c) Find the producer surplus.

\vskip 1.4in

d) Give a rough sketch of the supply and demand curves, indicating the
equilibrium price and quantity, as well as the regions whose area
represent the consumer and producer surplus.


\vfil\eject\noindent
12. (10 pts.) An employee of a company is offered a choice between the
following two retirement plans. For the first plan the company will open
an IRA with initial amount of \$6,000, and thereafter for the next 25
years an income stream of \$15,000 will be deposited in the IRA with an
annual rate of 8\% compounded continuously. For the second plan the
company will pay the employee \$1,200,000 at the end of 25 years. Which
one is the most beneficial plan for the employee? Explain your answer.


\vfil\eject\noindent
13. (20 pts.) The population of a developing region is growing at an
annual rate of \break 4.5\%. In other words, the growth rate is equal to 4.5\%
of the population at any given time. In 1995 the population of the
region was 5,000,000.

\vskip .2in 

a) Write a differential equation describing the population as a function
of time.

\vskip 1.7in

b) State an initial condition based on the given information. 

\vskip 1.7in

c) Solve the initial value problem describing the population function.

\vskip 1.7in

d) Find the population in 2005.}

\end
